Toronto won its eighth consecutive home game and swept a three-game series from the Rangers for the first time since August 3-5, 2000.
The 39-year-old Stairs went 3-for-3, scored once and recorded his second stolen base of the season. In the process, he became the third Canadian-born player to play in 1,500 career games, joining Larry Walker and Terry Puhl.
McGowan (8-5) allowed one run and eight hits over eight-plus innings, walking one and striking out six.
Despite being on the losing end, the Rangers' Catalanotto was still pleased for McGowan, his teammate in 2005 and 2006.
"We were all waiting for him to really blossom and this year he has," Catalanotto said. "I'm happy for him and he's doing a great job.
"He's finally put it all together," Catalanotto added. "We always knew he had the stuff but now he's putting it together, mentally too, and being able to execute."
Jeremy Accardo closed it out for his 20th save in 24 chances.
Texas lost its fourth straight, and has lost seven of nine following a season-high four-game winning streak. The Rangers head home after going 2-7 on a nine-game road trip to Kansas City, Cleveland and Toronto.
"It's been a real rough road trip," Catalanotto said. "We're happy to get home."
Catalanotto dented McGowan's line early, homering to lead off the game. It was his first leadoff homer of the season and Texas' third.
"Cat tried to make my confidence go down right from the get-go," McGowan said. "You've got to give credit to him. I thought I made a pretty good pitch."
Brandon McCarthy (5-8) allowed two runs and six hits over six innings to lose for the first time in three starts. He walked two and struck out four.
"He got us to the seventh and was in the game, but McGowan was just a little better," Texas manager Ron Washington said. "We're impressed with what he gave us today. We just didn't put enough runs on the board."
Toronto tied it in the bottom of the first when Frank Thomas' two-out RBI single scored Stairs, who had walked.
The Blue Jays got leadoff doubles from Gregg Zaun in the second and Stairs in the third, but couldn't advance the runner either time.
"He's a good professional hitter," McCarthy said of Stairs. "If you give him a pitch in the zone when he's ahead in the count, you're actually not expecting to get the ball back. He got me in a couple of big situations that I'd love to go back and do over again."
Texas loaded the bases with two out in the seventh, but McGowan got Ian Kinsler to ground into an inning-ending fielder's choice.
"It was a big shutdown inning," McGowan said.
- The Blue Jays have outscored opponents by a 54-15 margin over their eight-game home winning streak. Toronto's last home defeat was a 4-2 loss to Seattle on July 20.
- Toronto and Texas split the 10-game season series, ending the Rangers' five-year winning streak in head-to-head matchups.
- Several members of the Los Angeles Galaxy soccer team, including David Beckham, will attend Monday afternoon's game between the Blue Jays and the New York Yankees. The Galaxy's Canadian-born coach, Frank Yallop, is scheduled to throw out the first pitch.